On the Witt Vector Frobenius

Home , Frobenius endomorphism, Witt vector

ON THE WITT VECTOR FROBENIUS

CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA

Abstract. We study the kernel and cokernel of the Frobenius map on the p-typical Witt vectors of a commutative ring, not necessarily of characteristic p. We give many equivalent conditions to surjectivity of the Frobenus map on both finite and infinite length Witt vectors. In particular, surjectivity on finite Witt vectors turns out to be stable under certain integral extensions; this provides a clean formulation of a strong generalization of Faltings’s almost purity theorem from p-adic Hodge theory, incorporating recent improvements by Kedlaya–Liu and by Scholze.

Introduction Fix a prime number p. To each ring R (always assumed commutative and with unit), we may associate in a functorial manner the ring of p-typical Witt vectors over R, denoted W (R), and an endomorphism F of W (R) called the Frobenius endomorphism. The ring W (R) is set-theoretically an infinite product of copies of R, but with an exotic ring structure when p is not a unit in R; for example, for R a perfect ring of characteristic p, W (R) is the unique strict p-ring with ∼ W (R)/pW (R) = R. In particular, for R = Fp, W (R) = Zp. In this paper, we study the kernel and cokernel of the Frobenius endomorphism on W (R). In case p = 0 in R, this map is induced by functoriality from the Frobenius endomorphism of R, and in particular is injective when R is reduced and bijective when R is perfect. If p 6= 0 in R, the Frobenius map is somewhat more mysterious. To begin with, it is never injective; it is easy to construct many elements of the kernel. On the other hand, Frobenius is surjective in some cases, although these seem to be somewhat artificial; the simplest nontrivial example we have found is the valuation subring of a spherical completion of Qp. While surjectivity of Frobenius on full Witt vectors is rather rare, some weaker conditions turn out to be more relevant to applications. For instance, one can view the full ring of Witt vectors as an inverse limit of finite-length truncations, and surjectivity of Frobenius on finite levels is satisfied quite often. For instance, this holds for R equal to the ring of integers in any infinite algebraic extension of Q which is sufficiently ramified at p (e.g., the p-cyclotomic extension). In fact, this condition can be used to give a purely ring-theoretic formulation of a very strong generalization of Faltings’s almost purity theorem [2]. The theorem of Faltings features prominently in the theory of comparison isomorphisms in p-adic Hodge

Date: October 1, 2012. 2010 Mathematics Subject Classiﬁcation. Primary 13F35. Davis was supported by the Max-Planck-Institut f¨urMathematik (Bonn). Kedlaya was supported by NSF (CAREER grant DMS-0545904, grant DMS-1101343), DARPA (grant HR0011- 09-1-0048), MIT (NEC Fund, Green professorship), UCSD (Warschawski professorship).

1 2 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA theory; the generalization in question emerged recently from work of the second author and Liu [9] and of Scholze [11]. One principal motivation for studying the Frobenius on Witt vectors is to reframe p-adic Hodge theory in terms of Witt vectors of characteristic 0 rings, and ultimately to globalize the constructions with an eye towards study of global ´etalecohomology, K-theory, and L-functions. We will pursue these goals in subsequent papers.

1. Witt vectors Throughout this section, let R denote an arbitrary commutative ring. For more details on the construction of p-typical Witt vectors, see [7, Section 0.1] or [5, Section 17]; the latter treats big Witt vectors as well as p-typical Witt vectors.

Definition 1.1. For each nonnegative integer n, the ring Wpn (R) is defined to n+1 have underlying set Wpn (R) := R with an exotic ring structure characterized by functoriality in R and the property that for i = 0, . . . , n, the pi-th ghost component map wpi : Wpn (R) → R defined by pi pi−1 i wpi (r1, rp, . . . , rpn ) = r1 + prp + ··· + p rpi is a ring homomorphism. These rings carry Frobenius homomorphisms

F : Wpn+1 (R) → Wpn (R), again functorial in R, such that for i = 0, . . . , n, we have wpi ◦F = wpi+1 . Moreover, there are additive Verschiebung maps V : Wpn (R) → Wpn+1 (R) defined by the formula V (r1, . . . , rpn ) = (0, r1, . . . , rpn ). There is a natural restriction map Wpn+1 (R) → Wpn (R) obtained by forgetting the last component; define W (R) to be the inverse limit of the Wpn (R) via these restriction maps. The Frobenius homomorphisms at finite levels then collate to define another Frobenius homomorphism F : W (R) → W (R); there is also a collated Verschiebung map V : W (R) → W (R). The ghost component maps also collate to define a ghost map: w : W (R) → RN. We equip the target with component-wise ring operations; the map w is then a ring homomorphism. In either Wpn (R) or W (R), an element of the form (r, 0, 0,... ) is called a Teichmüller element and denoted [r]. These elements are multiplicative: for all r1, r2 ∈ R,[r1r2] = [r1][r2]. We will need the following properties of Witt vectors for R arbitrary (not necessarily of characteristic p). See [6, Lemma 1.5]. Here and in what follows, we write x for a Witt vector with components x1, xp,... . (a) For r ∈ R, F ([r]) = [rp]. (b) For x ∈ Wpn (R), (F ◦ V )(x) = px. (c) For x ∈ Wpn (R) and y ∈ Wpn+1 (R), V (xF (y)) = V (x)y. Pn i (d) For x ∈ Wpn (R), x = i=0 V ([xpi ]). Remark 1.2. A standard method of proving identities about Witt vectors and their operations is reduction to the universal case: take R to be a polynomial ring in many variables over Z, form Witt vectors whose components are distinct variables, then verify the desired identities at the level of ghost components. This suffices because R is now p-torsion-free, so the ghost map is injective. We will need a couple of other p-divisibility properties. We first prove the following lemma. ON THE WITT VECTOR FROBENIUS 3

Lemma 1.3. In W (Z/p2Z), we have p = (p, (−1)p−1, 0, 0,... ) = [p] + V ([(−1)p−1]). p p−1 Proof. Write p = (x1, xp,... ) ∈ W (Z). Then x1 = p and xp = (p−p )/p = 1−p , which is congruent to 1 mod p2 if p > 2 and to 3 mod 4 if p = 2. We now show by 2 induction on n that for each n ≥ 1, we have xpi ≡ 0 mod p for 2 ≤ i ≤ n. The base case n = 1 is vacuously true. For the induction step, considering the pn-th ghost component of p, write n−1 n pn pn−1 X i pn−i p xpn = −x1 + p(1 − xp ) − p xpi . i=2 To complete the induction, it suffices to check that each term on the right side has p-adic valuation at least n + 2. This is clear for the first term because pn ≥ n + 2. p−1 p−1 p 3 For the second term we have xp ≡ (−1) mod p and so xp ≡ 1 mod p (we pn−1 3+n−2 treat p = 2 and p > 2 separately). We then have xp ≡ 1 mod p , so the second term is indeed divisible by pn+2. For the terms in the sum, the claim is n−i again clear because i + 2p ≥ i + 2(n − i + 1) ≥ n + 2. Lemma 1.4. Take x, y ∈ W (R) with F (x) = y. p (a) For each nonnegative integer i, we have ypi = xpi +pxpi+1 +pfpi (x1, . . . , xpi ) where fpi is a certain universal polynomial with coefficients in Z which is i+1 j homogeneous of degree p for the weighting in which xpj has weight p . pi+1 (b) For i ≥ 1, the coefficient of x1 in fpi equals 0. pi (c) For i ≥ 2, the coefficient of xp in fpi is divisible by p. p p−2 (d) The coefficient of xp in fp equals −p modulo p. (e) For p = 2 and i ≥ 2, f2i belongs to the ideal generated by the elements 2 2, x1, x2 − x4, x8, . . . , x2i .

Proof. By reduction to the universal case, we see that ypi equals a universal polyno- i+1 mial in x1, . . . , xpi+1 with coefficients in Z which is homogeneous of degree p for p the given weighting. This polynomial is congruent to xpi modulo p by [7, (1.3.5)]. Each of the remaining assertions concerns a particular coefficient of this polynomial, and so may be checked after setting all other variables to 0. To finish checking (a), we must check that fpi does not depend on xpi+1 . We assume x1 = ··· = xpi = 0, i+1 i so that x = V ([xpi+1 ]); then F (x) = pV ([xpi+1 ]) = (0,..., 0, pxpi+1 ). To check (b), we may assume that xp = xp2 = ··· = xpi = 0, so that x = [x1]. p In this case, F (x) = [x1], so the claim follows. To check (c), we may assume that x1 = xp2 = xp3 = ··· = xpi = 0, so that x = V ([xp]). In this case, the claim is 2 that ypi ≡ 0 mod p for i ≥ 2. Since F (x) = p[xp], by homogeneity it is sufficient to check the claim for xp = 1. In this case, it follows from Lemma 1.3. We may similarly check (d). To check (e), we may assume that x1 = x8 = x16 = ··· = 0, and so we have 2 x = V ([x2]) + V ([x4]). By homogeneity, it is sufficient to check the claim in the case x2 = x4 = 1. In W (Z), we have V (1) = 1 + [−1] by computation of ghost components, so 1 + V (1) = 2 + [−1]. In W (Z/4Z), by Lemma 1.3 we have F (x) = 2 + 2V (1) = [2] + V ([−1]) + 2V (1) = [2] + V ([−1] + 2) = [2] + V (1) + V 2(1). This implies the desired result. 4 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA

Remark 1.5. Suppose R is a ring in which p = 0, and let ϕ : R → R denote the Frobenius homomorphism on R. Then applying Lemma 1.4, we have p p p F (r1, rp, rp2 ,...) = (r1 , rp, rp2 ,...). As a result, F is injective/surjective/bijective if and only if ϕ is injective/surjective/bijective. In particular, F is injective if and only if R is reduced, and F is bijective if and only if R is perfect. Similarly, the ﬁ- p p p nite level Frobenius map, which sends (r1, . . . , rpn ) to (r1 , rp, . . . , rpn−1 ), is injective only if R = 0, and is surjective if and only if ϕ is surjective.

2. The kernel of Frobenius When R is a ring not of characteristic p, then F : W (R) → W (R) cannot be injective; for instance, the Cartier-Dieudonné-Dwork lemma [5, Lemma 17.6.1] implies that p, 0, 0,... arises as the sequence of ghost components of some element of W (Z). More generally, one can determine exactly which elements of R can occur as the first component of an element of the kernel of F . This will be useful in our analysis of surjectivity of F . p Definition 2.1. Given a ring R, define sets I0 := R and Ii := {r ∈ R | r ∈ pIi−1} for i > 0; it is apparent that I0 ⊇ I1 ⊇ I2 ⊇ · · · . We will see below that each Ii is ∞ p an ideal. Also define I∞ = ∩i=1Ii, so that I∞ = {r ∈ R | r ∈ pIi for all i ≥ 1}.

Lemma 2.2. For each i ≥ 0, the set Ii deﬁned above is an ideal.

Proof. We proceed by induction on i, the case i = 0 being obvious. Given that Ii−1 is an ideal, it is clear that Ii is closed under multiplication by arbitrary elements of R. It remains to show that if x, y ∈ Ii, then x+y ∈ Ii. Using the deﬁnition, we must p p−1 p−1 p p p check that x +px y+···+pxy +y ∈ pIi−1. That x , y ∈ pIi−1 follows from x, y ∈ Ii. That the remaining terms are in pIi−1 follows from x, y ∈ Ii ⊆ Ii−1. Remark 2.3. Suppose that R is a valuation ring with valuation v, p is nonzero in R, and v(p) is p-divisible in the value group of v. Then for each nonnegative integer n, 1 1 the ideal In consists of all x ∈ R such that v(x) ≥ Nn for Nn = p + ··· + pn v(p). In particular, In is principal, generated by any x ∈ R for which v(x) = Nn. If 1 moreover v is a real valuation and there exists y ∈ R such that v(y) = p−1 v(p), then I∞ is the principal ideal generated by y. A typical example would be the ring of integers in an algebraic closure of Qp or in the completion thereof.

Deﬁnition 2.4. For any ring R, any r0 ∈ R, and any i ≥ 0 (including i = ∞), deﬁne B(r0,Ii) := r0 + Ii = {r ∈ R | r − r0 ∈ Ii}. The notation is meant to suggest that B is a ball centered at r0.

The signiﬁcance of the ideals Ii is the following. Proposition 2.5. Let R be a ring, let i be a positive integer, and let n be either 0 ∞ or an integer greater than or equal to i. For x, y ∈ Wpi (R), put x := F (x), y0 := F (y). 0 0 (a) If xpj − ypj ∈ In−j for j = 0, . . . , i, then xpj − ypj ∈ pIn−j−1 for j = 0, . . . , i − 1. 0 0 (b) If xpj − ypj ∈ pIn−j for j = 0, . . . , i and xpi − ypi ∈ In−i, then xpj − ypj ∈ In−j for j = 0, . . . , i. In particular, if F (x) = F (y), then xpj − ypj ∈ Ii−j for j = 0, . . . , i − 1. ON THE WITT VECTOR FROBENIUS 5

(c) Choose x1, . . . , xpi−1 , y1, . . . , ypi−1 ∈ R with xpj − ypj ∈ In−j for j = 0, . . . , i − 1. Assume also that F (x1, xp, . . . , xpi−1 ) = F (y1, yp, . . . , ypi−1 ) if i > 1. Then for any ypi ∈ R, there exists xpi ∈ B(ypi ,In−i) such that F (x) = F (y). (d) For any x ∈ W (R) for which xpi ∈ pI∞ for all i, there exists y ∈ W (R) for which y1 = 0, ypi ∈ I∞ for all i, and F (y) = x. 0 0 p p Proof. To check (a), apply Lemma 1.4(a) to write xpj − ypj = xpj − ypj + p(xpj+1 − ypj+1 ) + p(fpj (x1, . . . , xpj ) − fpj (y1, . . . , ypj )). Writing ypj = xpj − (xpj − ypj ), we p p p note that xpj − ypj belongs to the ideal generated by (xpj − ypj ) and p(xpj − ypj ). Note also that p(fpj (x1, . . . , xpj ) − fpj (y1, . . . , ypj )) belongs to the ideal generated 0 0 by p(x1 − y1), . . . , p(xpj − ypj ). It follows that xpj − ypj ∈ pIn−j−1. To check (b), we ﬁrst check that under the hypotheses of (b), if there exists 0 ≤ k ≤ n − i + 1 such that xpj − ypj ∈ Ik for j = 0, . . . , i, then xpj − ypj ∈ Ik+1 for j = 0, . . . , i − 1. For j ∈ {0, . . . , i − 1}, apply Lemma 1.4(a) to write p 0 0 p p p (xpj − ypj ) − (xpj − ypj ) = ((xpj − ypj ) − xpj + ypj )

− p(xpj+1 − ypj+1 + fpj (x1, . . . , xpj ) − fpj (y1, . . . , ypj )). p 0 0 From this equality we see that (xpj −ypj ) −(xpj −ypj ) belongs to the ideal generated by p(x1 − y1), . . . , p(xpj+1 − ypj+1 ). This ideal is contained in pIk by hypothesis. 0 0 We also have xpj − ypj ∈ pIn−j, and because n − j ≥ n − i + 1 ≥ k, we have 0 0 p xpj − ypj ∈ pIk as well. Hence (xpj − ypj ) ∈ pIk, and so we have xpj − ypj ∈ Ik+1 as claimed. Note that the hypothesis of the previous paragraph is always satisfied for k = 0 because I0 = R. This gives us control over the terms x1 −y1, . . . , xpi−1 −ypi−1 . Since xpi −ypi ∈ In−i by assumption, we may induct on k to deduce that xpj −ypj ∈ In−i+1 for j = 0, . . . , i − 1. In particular, xpi−1 − ypi−1 ∈ In−i+1; we may now induct on i to deduce (b). To check (c), by Lemma 1.4(a) again, it suffices to find xpi ∈ B(ypi ,In−i) such p p that xpi−1 +pxpi +pfpi−1 (x1, . . . , xpi−1 ) = ypi−1 +pypi +pfpi−1 (y1, . . . , ypi−1 ). Note that xpi−1 − ypi−1 ∈ In−i+1 and x1 − y1, . . . , xpi−1 − ypi−1 ∈ In−i, so as in the proof p p of (a) we have xpi−1 − ypi−1 + p(fpi−1 (x1, . . . , xpi−1 ) − fpi−1 (y1, . . . , ypi−1 )) ∈ pIn−i. To check (d), we construct the ypi recursively, choosing y1 = 0. Given y1, . . . , ypi , p we must choose ypi+1 so that in the notation of Lemma 1.4(a), we have ypi +pypi+1 + p pfpi (y1, . . . , ypi ) = xpi . This is possible because ypi , pfpi (y1, . . . , ypi ), and xpi all belong to pI∞. Corollary 2.6. Let R be a ring and let n be either ∞ or a positive integer. Then an element r ∈ R occurs as the first component of an element of the kernel of F : Wpn (R) → Wpn−1 (R) if and only if r ∈ In.

Proof. Suppose that n < ∞. If r = z1 for z ∈ Wpn (R) such that F (z) = 0, then trivially zpn ∈ I0. By Proposition 2.5(b), z1 ∈ In; the same conclusion holds for n = ∞. Conversely, suppose r ∈ In. Put z1 = r. By Proposition 2.5(c) applied repeatedly, for each positive integer i ≤ n, we can find zpi ∈ In−i so that F (z1, zp, . . . , zpi ) = 0. This proves the claim. Remark 2.7. The image under the ghost map of any element in the kernel of F has the form (∗, 0, 0,...). If R is a p-torsion-free ring, the ghost map is injective, so any 6 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA element of the kernel of F is uniquely determined by its first component. In this case, we may combine Proposition 2.5(b) and (c) to deduce that if z ∈ Wpi (R) is such that F (z) = 0 and z1 ∈ In for some n ≥ i, then zpj ∈ In−j for j = 0, . . . , i. 3. Surjectivity conditions Surjectivity of the Witt vector Frobenius turns out to be a subtler property than injectivity, because there are many partial forms of surjectivity which occur much more frequently than full surjectivity. We first list a number of such conditions, then identify logical relationships among them. Definition 3.1. For R an arbitrary ring, label the conditions on R as follows. (i) F : W (R) → W (R) is surjective. (ii) F : Wpn (R) → Wpn−1 (R) is surjective for all n ≥ 2. 0 (ii) F : Wp2 (R) → Wp(R) is surjective. (iii) For every x ∈ W (R), there exists r ∈ R such that x − [r] ∈ pW (R). (iv) The image of F : W (R) → W (R) contains all Teichmüllerelements [r]. (v) F : Wpn (R) → Wpn−1 (R) contains all elements of the form (r, 0,..., 0) for all n ≥ 2. 0 (v) F : Wp2 (R) → Wp(R) contains all elements of the form (r, 0). (vi) The image of F : W (R) → W (R) contains V (1). (vii) For all n ≥ 2, the image of F : Wpn (R) → Wpn−1 (R) contains V (1). n−1 (viii) For all n ≥ 2, the image of F : Wpn (R) → Wpn−1 (R) contains V (1). (ix) The image of F : Wp2 (R) → Wp(R) contains V (1). n (x) F : Wpn (R) → W1(R) is surjective for all n ≥ 1. 0 (x) F : Wp(R) → W1(R) is surjective. (xi) R contains p−1. (xii) For any r0, r1, · · · ∈ R such that B(r0,I0) ⊇ B(r1,I1) ⊇ · · · (in the notation

of Definition 2.4), the intersection ∩i∈NB(ri,Ii) is non-empty. (xiii) The p-th power map on R/pI∞ (which need not be a ring homomorphism) is surjective. (xiv) For each n ≥ 1, the p-th power map on R/pIn is surjective. 0 (xiv) The p-th power map on R/pI1 is surjective. (xv) For every r ∈ R, there exists s ∈ R such that sp ≡ pr mod p2R. p (xvi) There exist r, s in R such that r ≡ −p mod psR and s ∈ I1. (xvii) There exist r, s in R such that rp ≡ −p mod psR and sN ∈ pR for some integer N > 0. (xviii) The Frobenius homomorphism ϕ : r 7→ rp on R/pR is surjective. These conditions are represented graphically in Figure 1. The conditions in the top left quadrant refer to infinite Witt vectors, those in the top right quadrant refer to finite Witt vectors, and those below the dashed line refer to R itself. Theorem 3.2. For any ring R, we have (ii) ⇔ (ii)0, (v) ⇔ (v)0, (x) ⇔ (x)0, and (xiv) ⇔ (xiv)0. In addition, each solid single arrow in Figure 1 represents a direct implication, and for each other arrow type, the conditions at the sources of the arrows of that type together imply the condition at the target. The proof of Theorem 3.2 will occupy most of the rest of this section. First, however, we mention some consequences of Theorem 3.2 and some negative results which follow from some examples considered in Section 4. ON THE WITT VECTOR FROBENIUS 7

' (i)t| (iv)/ / (v)o (ii)/ /7 Ya_g G G O U

# # (iii) (vi)i (vii) (viii)O (ix)/ O (x)O

(xi) (xii) (xiii) (xiv)/ (xv) (xvi)/ (xvii) (xviii)

Figure 1. Logical implications among conditions on the ring R.

Corollary 3.3. For any ring R, we have the following equivalences. • (i) ⇔ (ii) + (xii) ⇔ (iii) + (iv) ⇔ (iii) + (xiii) ( ) ( ) (vi), (vii), (viii), (ix), • (ii) ⇔ (v) ⇔ (xiv) ⇔ (x) or (xviii) + (xv), (xvi), or (xvii)

The equivalence (ii) ⇔ (xviii) + (xvii) will be needed later in the proof of Theorem 5.2. Remark 3.4. The following implications fail to hold by virtue of the indicated examples. • (i) 6⇒ (xi) by Example 4.7. • (ii) 6⇒ (i) by Example 4.4 (or from (ii) 6⇒ (iv) below). • (ii) 6⇒ (iii) by Example 4.4. • (ii) 6⇒ (iv) by Example 4.9. • (ii) 6⇒ (xii) by Example 4.4. • (iv) 6⇒ (i) by Example 4.4. • (vi) 6⇒ (xv) by Example 4.8. • (vi) 6⇒ (xviii) by Example 4.8. • (xii) 6⇒ (xvii) by Example 4.2. • (xv) 6⇒ (xviii) by Example 4.3. • (xviii) 6⇒ (xvii) by Example 4.2. Proof of Theorem 3.2. We now prove the implications represented in Figure 1. • (i) ⇒ (iv); (ii) ⇒ (ii)0;(ii) ⇒ (v); (ii) ⇒ (x); (ii)0 ⇒ (v)0;(iv) ⇒ (v); (v) ⇒ (v)0;(vi) ⇒ (vii); (vii) ⇒ (ix); (viii) ⇒ (ix); (x) ⇒ (x)0;(xiv) ⇒ (xiv)0;(xvi) ⇒ (xvii) Proof. These are all obvious. • (i) ⇒ (iii) P i Proof. Let x ∈ W (R) be arbitrary. We may write x = V ([xpi ]), and p because F ◦ V = p, we have F (x) ≡ [x1] mod pW (R). Since we are assuming that F is surjective, we deduce (iii). • (i) ⇒ (xii)

Proof. Fix elements ri as in condition (xii). Our strategy is to deﬁne an element y ∈ W (R) so that if x ∈ W (R) is such that F (x) = y, then we must 8 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA

∞ have x1 ∈ ∩i=0B(ri,Ii). To prescribe our element y ∈ W (R), it suffices to (pi) define compatible finite length Witt vectors y ∈ Wpi (R) for every i. (p) (p) Define x ∈ Wp(R) by x = (r1, 0) (the second component does not matter). Set y(1) := F (x(p)). Now inductively assume we have defined i (pi) (p ) x ∈ Wpi (R) for some i ≥ 1 and with first component x1 = ri. By i (pi) (p ) Proposition 2.5(b,c), we can find an element z ∈ Wpi (R) with z1 = (pi) (pi+1) ri+1 − ri ∈ Ii and with F (z ) = 0. Choose any x ∈ Wpi+1 (R) (pi) (pi) (pi) (pi) which restricts to x + z ∈ Wpi (R). Set y = F (x ). Then by (pi) construction our elements y yield an element of lim W i (R) ∼ W (R), ←− p = which we call y. By (i), we can find an element x such that F (x) = y. Because F (x) and i+1 (pi+1) (p ) F (x ) have the same initial i+1 components, we have that x1 ≡ x1 mod Ii+1 by Proposition 2.5(c). Because x1 does not depend on i, and i+1 (p ) ∞ x1 = ri+1, we have that x1 ∈ ∩i=0B(ri+1,Ii+1), as desired. • (ii) + (xii) ⇒ (i) Proof. Choose any y ∈ W (R). We will construct x ∈ W (R) such that F (x) = y. We use (ii) to find elements x(1), x(p),... ∈ W (R) so that (1) (p) F (x ) = (y1, ∗, ∗, ··· ),F (x ) = (y1, yp, ∗, ∗, ··· ), and so on. By (xii) and Proposition 2.5(b), we may choose xfpj which is in the intersection (pj ) (pj+1)) B0(xpj ,I0) ∩ B1(xpj ,I1) ∩ · · · . Put ye := F (xf1, xfp,... ). We first apply (pk) (pk) Proposition 2.5(a) to (xf1,..., x]pi+1 ) and (x1 , . . . , xpi+1 ) for fixed i and increasing k, which implies that yfpi −ypi ∈ pI∞ for each nonnegative integer i. This means that y and ye have the same image in W (R/pI∞), so the difference z = y−ye has all of its components in pI∞. By Proposition 2.5(d), z is in the image of F , as then is y. • (iii) + (iv) ⇒ (i) Proof. This is obvious, given that any element px0 = F (V (x0)) is in the image of Frobenius. • (iv) ⇒ (xiii); (v) ⇒ (xiv); (v)0 ⇒ (xiv)0

Proof. Suppose that n ≥ 2 and that x ∈ Wpn (R) and r ∈ R are such that F (x) = [r]. For each of k = 0, . . . , n − 1, we check that xp, xp2 , . . . , xpn−k belong to Ik. This is clear for k = 0. Given the claim for some k < n − 1, for i = 1, . . . , n − 1 − k we may apply Lemma 1.4(a) to deduce p that xpi + pxpi+1 + pfpi (x1, ..., xpi ) = 0. By Lemma 1.4(b), fpi contains no pi+1 pure power of x1 , so fpi (x1, . . . , xpi ) belongs to the ideal generated by xp, . . . , xpi . Therefore −pxpi+1 and −pfpi (x1, . . . , xpi ) belong to pIk, and so xpi ∈ Ik+1. This proves the claim. Consequently, xp ∈ In−1, and so p r − x1 = pxp ∈ pIn−1. The stated implications now follow. • (ix) ⇒ (xvi) Proof. We are assuming that we can ﬁnd x such that F (x) = V ([1]). Then p the ghost components of x must be (∗, 0, p). In other words, x1 + pxp = 0 2 p p 2 p and x1 + pxp + p xp2 = p. The ﬁrst equality tells us that x1 ∈ pR (and ON THE WITT VECTOR FROBENIUS 9

2 p p p 2 hence x1 ∈ p R). The second equality now tells us pxp ≡ p mod p R p p p and so xp ≡ 1 mod pR. By the binomial theorem, xp − 1 − (xp − 1) is in p pR, so if we put s := xp − 1, we have s ∈ pR. Returning to the equation p p p x1 + pxp = 0, we have x1 ≡ −p mod psR with s ∈ pR, as required. • (x)0 ⇒ (xviii); (xi) ⇒ (i) Proof. Working with ghost components as above, these are obvious. • (xiii) ⇒ (iv)

Proof. Given r ∈ R, by (xiii) we may choose x1 ∈ R, xp ∈ I∞ for which p r = x1 + pxp. We now show that we can choose xp2 , xp3 , · · · ∈ I∞ so that F (x1, . . . , xpn ) = (r, 0,..., 0) for each n ≥ 1. Given x1, . . . , xpn , define fpn as in Lemma 1.4(a). By Lemma 1.4(b), pn+1 fpn contains no pure power of x1 , so fpn (x1, . . . , xpn ) belongs to the ideal generated by xp, . . . , xpn , which by construction is contained in I∞. p It follows that −xpn − fpn (x1, . . . , xpn ) ∈ pI∞, so we can find xpn+1 ∈ pI∞ p for which xpn +pxpn+1 +fpn (x1, . . . , xpn ) = 0. By Lemma 1.4(a), this choice of xpn+1 has the desired effect. • (xv) ⇒ (vi)

Proof. We will produce elements x1, xp,... of R such that F (x1, xp,...) = (0, 1, 0, 0,...) = V (1). Using (xv), choose r so that rp ≡ −p mod p2. Set x1 := r. Then clearly we can choose xp ≡ 1 mod p such that F (x1, xp) = (0). Next, in the notation of Lemma 1.4(a), we wish to choose xp2 so that p xp + pxp2 + pfp(x1, xp) = 1.

We also wish to ensure that if p > 2, then xp2 ≡ 0 mod p, while if p = p 2, then xp2 = 1 mod p. To see that this is possible, we note xp ≡ 1 2 mod p . We then observe that fp(x1, xp) consists of an element of the ideal p p generated by x1 (which is a multiple of p) plus some constant times xp. By Lemma 1.4(c,d), when p > 2 this constant is divisible by p, so pfp(x1, xp) ≡ 2 2 0 mod p . If p = 2, this constant is 1 mod 2, so pfp(x1, xp) ≡ 2 mod p . In either case, we obtain xp2 of the desired form. Now assume that for some i ≥ 2, we have found x1, xp, . . . , xpi such that xpj ≡ 0 mod p for j ≥ 3 and such that F (x1, xp, . . . , xpi ) = (0, 1, 0,..., 0). We claim that we can ﬁnd xpi+1 ≡ 0 mod p with F (x1, xp, . . . , xpi+1 ) = (0, 1, 0,..., 0). We claim we can ﬁnd xpi+1 ≡ 0 mod p such that p xpi + pxpi+1 + pfpi (x1, . . . , xp) = 0

and with fpi (x1, . . . , xp) ≡ 0 mod p. This follows by Lemma 1.4(c,e). • (xv) ⇒ (viii)

Proof. Our goal is to find an element x = (x1, xp, . . . , xpn ) such that F (x) = n−1 V (1). First, set x^pn−1 = 1, and then find x^pn−2 ,..., xf1 (in that order) p 2 such that xpi ≡ −px]pi+1 mod p R. This is possible by (xv). Note that p f xfpi ∈ pR for 0 ≤ i ≤ n − 2. We next construct the elements x1, . . . , xpn−1 . Set x1 := xf1. Assume we have found x1, . . . , xpi with xpj ≡ xfpj mod pR for some i ≤ n − 2. Using the notation of Lemma 1.4(a), we first must choose xpi+1 which satisfies 10 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA

p xpi + pxpi+1 + pfpi (x1, . . . , xpi ) = 0. Write xpi+1 = x]pi+1 + pypi+1 . We must choose ypi+1 so that p 2 xpi + px]pi+1 + p ypi+1 + pfpi (x1, . . . , xpi ) = 0. p 2 Because xfpi + px]pi+1 ≡ 0 mod p and xfpi ≡ xpi mod p, we deduce that p 2 xpi + px]pi+1 ≡ 0 mod p . We further have that pfpi (x1, . . . , xpi ) ≡ 0 mod p2; this follows from the homogeneity result in Lemma 1.4(a) and p the fact that xpj ≡ 0 mod p for all j. This shows that we can ﬁnd the required ypi+1 . Finding the last component xpn is a little diﬀerent, because the last component of V n−1(1) is 1 instead of 0. This means that we need p xpn−1 + pxpn + pfpn−1 (x1, . . . , xpn−1 ) = 1.

But this is easy, because we know xpn−1 ≡ 1 mod p. • (xvi) ⇒ (ix) p Proof. By Lemma 1.4(a), we must find x1, xp, xp2 such that x1 + pxp = 0 p and xp + pxp2 + pfp(x1, xp) = 1. By (xvi), we can find an element x1 such p p that x1 + p + psr = 0 where s ∈ pR; we then choose xp = 1 + sr. It’s then p clear that xp + pfp(x1, xp) ≡ 1 mod pR, and so we can find xp2 forcing p xp + pxp2 + pfp(x1, xp) = 1, as desired. • (xviii) ⇒ (x) n−i Pn i p Proof. For any r ∈ R, we must find r1, . . . , rpn such that i=0 p rpi = r. pn We first find r1, s such that r − r1 = ps by repeatedly applying (xviii). To find the remaining rpi , we apply the induction hypothesis to s. • (x) ⇒ (xviii); (x)0 ⇒ (x) Proof. We have already seen (x)0 ⇒ (xviii). The two results follow because 0 we have also shown (x) ⇒ (x) and (xviii) ⇒ (x). • (x) + (xvii) ⇒ (xv) p Proof. By (xvii), we can find s1, s2 ∈ R for which s1 = −p(1 − s2) and N 0 s2 ∈ (p) for some N > 0. We have already seen that (x) ⇒ (x) ⇒ (xviii). p Given any r ∈ R, by (xviii) we can find s3 ∈ R with s3 ≡ −r(1 + s2 + N−1 N p ··· + s2 ) mod p. Since s2 ≡ 0 mod p, for s = s1s3 we have s = N−1 N 2 pr(1 − s2)(1 + s2 + ··· + s2 ) = pr(1 − s2 ) ≡ pr mod p . • (x) + (xvii) ⇒ (ii) Proof. We just saw that (x) + (xvii) ⇒ (xv), and we know (xv) ⇒ (viii). We will thus use (viii) freely below. We prove that F : Wpn (R) → Wpn−1 (R) is surjective for n ≥ 1 by induction on n. The base case n = 1 is exactly (x)0. Now assume the result for some fixed n − 1, pick any y ∈ Wpn (R), and consider the diagram

F 0 Wpn+1 (R) 3 r / y ∈ Wpn (R) y ∈ Wpn (R) res res res

' F ' W n (R) 3 s / y| . p Wpn−1 (R) ON THE WITT VECTOR FROBENIUS 11

The term s exists by our inductive hypothesis and the term r exists because restriction maps are surjective. If we had y = y0, we would be done. 0 0 n Find x ∈ Wpn+1 (R) with F (x ) = V ([1]) using (viii). Then, using (x), 00 0 n n+1 00 ﬁnd x ∈ Wpn+1 (R) with y − y = V (F (x )). A calculation now shows 0 00 F (r + x x ) = y, as desired. • (xiv)0 ⇒ (x) + (xvii); (xiv) ⇒ (xv) Proof. These are compositions of implications we have already proved. • (ii)0 ⇒ (ii); (v) ⇒ (ii); (v)0 ⇒ (v); (xiv) ⇒ (v); (xiv)0 ⇒ (xiv) Proof. We will prove that all six conditions appearing in the statement are equivalent. We have already proven the following implications: (ii) / (v) / (xiv)

(ii)0 / (v)0 / (xiv)0. Thus, it suffices to prove that (xiv)0 ⇒ (ii). This follows because we have 0 seen above that (xiv) ⇒ (x) + (xvii) ⇒ (ii). We conclude the discussion by making some additional observations in the case of valuation rings. Remark 3.5. Let R be a valuation ring with valuation v for which 0 < v(p) < +∞, and introduce the following new condition. (xix) There exists x ∈ R with 0 < v(x) < v(p). We then have (ii) ⇔ (xv) ⇔ (xviii) + (xix). Namely, by Theorem 3.2, it suffices to check that (xv) implies (xviii) and that the two conditions (xviii) + (xix) together imply (xvi). These implications are verified as follows. p 2 p Given (xv), for any x ∈ R we can find y1, y2 ∈ R with y1 ≡ px mod p R, y2 ≡ p 2 1 1 mod p R. Then v(y1) = p (v(p) + v(x)), v(y2) = p v(p), so z := y1/y2 is an element of R satisfying zp ≡ x mod pR. This yields (xviii). Given (xviii) + (xix), there exist y, z ∈ R with yp ≡ x mod p, zp ≡ p/x 1 1 mod pR. Since 0 < v(x), v(p/x) < v(p), we have v(y) = p v(x), v(z) = p (v(p) − 1 p v(x)), so v(yz) = p . Therefore, u := (yz) /p is a unit in R, so there exists w ∈ R such that wp ≡ −u−1 mod pR. Thus we have puwp ≡ −p mod p2R and (yzw)p ≡ −p mod p2R, yielding (xvi). (As a byproduct of the argument, we note that (ii) implies that v(p) is p-divisible.)

4. Examples We now describe some simple examples realizing distinct subsets of the conditions considered above. Example 4.1. Take R to be any ring in which p is invertible. Then by Theorem 3.2, all of our conditions hold.

Example 4.2. Take R = Z. In this case, Ii = (p) for all i ≥ 1. Thus (xvii) fails, and consequently, neither (i) nor (ii) holds for R = Z. On the other hand, (xii) does hold for R = Z. To see this, we must show that any descending chain of balls · · · ⊇ B(ri−1, (p)) ⊇ B(ri, (p)) ⊇ · · · has nonempty intersection, which is clear. 12 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA

Example 4.3. Take R = Fp[T ]. In this case, (xv) is satisﬁed trivially, because pr = 0 for all r ∈ R. On the other hand, (xviii) is not satisﬁed.

Example 4.4. Take R = OCp . Then (xiii) holds because R is integrally closed in the algebraically closed ﬁeld Cp; this implies that R satisﬁes (iv), (v), (vi), (vii), (viii), (ix), (x), (xiii), (xiv), (xv), (xvi), (xvii), (xviii). On the other hand, (xii) does not hold by Lemma 4.5, so R does not satisfy (i), (iii), (xi), (xii).

Lemma 4.5. The ring R = OCp does not satisfy (xii).

Proof. By Remark 2.3, for n a nonnegative integer, In is the principal ideal gen- 1 1 1 +···+ n erated by p p p , while I∞ is the principal ideal generated by p p−1 . Each ball B(r, I∞) contains an element which is algebraic over Q, since such elements are 0 dense in Cp by Krasner’s lemma. Furthermore, if two balls B(r, I∞) and B(r ,I∞) intersect, they are in fact equal. Therefore, there are only countably many such balls. On the other hand, one can construct uncountably many decreasing se- quences B(r0,I0) ⊇ B(r1,I1) ⊇ · · · no two of which have the same intersection.

For instance, take x0, x1,... to be Teichm¨ullerelements in W (Fp) ⊆ OCp , and put 1 1 1 p + 2 r0 = x0, r1 = r0 + x1p p , r2 = r1 + x2p p ,.... ∞ Then any two of the resulting intersections ∩i=0B(ri,Ii) are disjoint. Remark 4.6. It is possible to give a more constructive proof of Lemma 4.5 using the explicit description of OCp given in [8]. Example 4.7. Let R be a spherically complete valuation ring (i.e., any decreasing sequence of balls in R has nonempty intersection) such that the valuation of p is nonzero and p-divisible (so the In are as computed in Remark 2.3). Then R satisﬁes (xii).

In particular, let R denote the spherical completion of OCp constructed by Poo- nen in [10]. Namely, let Zp tQ denote the ring of generalized power series over P i p; its elements are formalJ sumsK cit with ci ∈ p such that the set Z i∈Q,i≥0 Z {i ∈ Q : ci 6= 0} is well-ordered. This ring is spherically complete for the t-adic ∼ valuation. We then take R = Zp tQ /(t − p), so that R/(p) = Fp tQ /(t). From J K J K this description, it is clear that R satisfies (xii) and (xviii); since R contains OCp , it also satisfies (xvi). Putting this together, we deduce that R satisfies (i). 2 Example 4.8. Take R = Z[µp2 ], where µp2 is a primitive (p )-nd root of unity. Pp−1 i Condition (vi) holds because if x = i=0 [µp2 ] ∈ W (R), then F (x) = V (1). (Since R is p-torsion-free, this last equality can be checked at the ghost component level, where it is apparent.) On the other hand, (xviii) does not hold: the element (1−ωp2 ) 1 has p-adic valuation p(p−1) , but there is no element of R which has p-adic valuation 1 p2(p−1) . Similarly, (xv) does not hold.

Example 4.9. Take R = Z[µp∞ ], i.e., the ring of integers in the maximal abelian extension of Q. We will see that (ii) holds but (iv) does not. (The same analysis applies to Zp[µp∞ ] or its p-adic completion.) Note that R satisfies (vi) because R contains the subring Z[µp2 ] which satisfies (vi) by Example 4.8. Thus to establish (ii), it is sufficient to check condition (xviii).

For this, note that for any expression a1µpi1 + ··· + anµpin with a1, . . . , an ∈ Z, we p have a1µpi1 + ··· + anµpin ≡ (a1µpi1+1 + ··· + anµpin+1 ) mod p. ON THE WITT VECTOR FROBENIUS 13

To establish that R does not satisfy (iv), we will check that R does not satisfy (xiii). We will do this assuming p > 2, by checking that the congruence xp ≡ 1 − p mod pI∞ has no solution. This breaks down for p = 2; in this case, one can show 2 by a similar argument that the congruence x ≡ i + 2µ8 mod 2I∞ has no solution. Assume by way of contradiction that p > 2 and there exists x ∈ R for which p x − 1 + p ∈ pI∞. Recall that by Remark 2.3, I∞ is the principal ideal generated 1/(p−1) by p . Choose an integer n ≥ 2 for which x ∈ Z[µpn ], and put z = 1 − µpn . n−1 n−2 n By an elementary calculation, zp −p ≡ −p (mod zp ); in particular, the p- 1 adic valuation of 1 − µpn is pn−1(p−1) . There must thus exist y ∈ Z[µpn ] such that n−1 n−2 n (1 + yzp −p )p ≡ 1 − p (mod zp ); subtracting 1 from both sides and dividing n−1 n−2 n−1 by p, we obtain the congruence yzp −p − yp ≡ −1 (mod zp ). Using the pn−1 ∼ pn−1 isomorphism Z[µpn ]/(z ) = Fp[T ]/(T ) sending z to T , we obtain a solution p pn−1−pn−2 pn−1 w of the congruence w −wT ≡ 1 (mod T ) in Fp[T ]. However, no such n−2 solution exists: such a solution would satisfy w 6≡ 0 (mod T p ), so there would be a largest index i < pn−2 such that the coeﬃcient of T i in w is nonzero. But n−1 n−2 n−1 n−2 then T i+p −p would appear with a nonzero coeﬃcient in wp − wT p −p .

5. Almost purity Let us say that a ring R is Witt-perfect if condition (ii) holds. We conclude with one motivation for studying Witt-perfect rings: they provide a natural context for the concept of almost purity, as introduced by Faltings [2] and studied more recently by the second author and Liu in [9] and by Scholze in [11]. More precisely, the Witt-perfect condition amounts to an absolute version (not relying on a valuation subring) of the condition for a ring to be integral perfectoid in the sense of Scholze. (For a valuation ring, another equivalent condition is to be strictly perfect in the sense of Fargues and Fontaine [3].) We begin by defining the adverb almost in this context. See [4] for a more general setting. Definition 5.1. A p-ideal of a ring R is an ideal I such that In ⊆ (p) for some positive integer n. An R-module M is almost zero if IM = 0 for every p-ideal I. Theorem 5.2. Let R be a p-torsion-free Witt-perfect ring which is integrally closed −1 in Rp := R[p ]. Let Sp be a finite étale Rp-algebra and let S be the integral closure of R in Sp. (a) The ring S is also Witt-perfect. (b) For any p-ideal I of R, there exist a finite free R-module F and R-module homomorphisms S → F → S whose composition is multiplication by some t ∈ R for which I ⊆ (t).

equal to the image of the natural map from HomR(S, R) to HomRp (Sp,Rp). In the language of almost ring theory, the conclusion here is that S is almost finite étale over R. Again, see [4] for detailed definitions.

Proof. For each t ∈ Q, choose integers r, s ∈ Z with s > 0 and r/s = t. Since R is integrally closed in Rp, the set −r s Rt := {x ∈ Rp : p x ∈ R} 14 CHRISTOPHER DAVIS AND KIRAN S. KEDLAYA depends only on t. The function v : Rp → (−∞, +∞] given by

v(x) := sup{t ∈ Q : x ∈ Rt} satisﬁes v(x − y) ≥ min{v(x), v(y)}, v(xy) ≥ v(x) + v(y), and v(x2) = 2v(x). −v(·) Let A be the separated completion of Rp under the norm |·| = e , and deﬁne the subring oA = {x ∈ A : |x| ≤ 1} and the ideal mA = {x ∈ A : |x| < 1}. Let −1 −1 ψ : Rp → A be the natural homomorphism; then R ⊆ ψ (oA) and ψ (mA) ⊂ R, −1 so ψ (oA)/R is an almost zero R-module. Since (ii) implies (xviii) and (xv), we can choose x1, x2 ∈ R with p 2 p x1 ≡ −p mod p R, x2 ≡ x1 mod pR.

Then ψ(x1), ψ(x2) are units in A, and for all y ∈ A, −1/p −1/p2 |ψ(x1)y| = p |y|, |ψ(x2)y| = p |y|. p −1 Given y ∈ oA/(p), choose y ∈ Rp so that ψ(y) lifts y. Then x2y ∈ ψ (mA) ⊂ R, p p so since R satisfies (ii), we can find z ∈ R with x2y ≡ z mod pR. The element p p ψ(z/x2) ∈ oA has the property that ψ(z/x2) ≡ ψ(y) mod (p/ψ(x2) )oA; it fol- p−1 p(p−1) p−1 lows that Frobenius is surjective on oA/(ψ(x1) ). Since ψ(x2) ≡ ψ(x1) mod poA, it also follows that Frobenius is surjective on oA/(p). That is, oA also satisfies (xviii); since (xvi) is evident (using x1), oA satisfies (ii).

Put B = A⊗Rp Sp and extend ψ by linearity to a homomorphism ψ : Sp → B. By [9, Theorem 3.6.12], there is a unique power-multiplicative norm on B under which it is a finite Banach A-module, and for this norm the subring oB = {x ∈ B : |x| ≤ 1} also satisfies (ii). As in [9, Remark 2.3.14], for mB = {x ∈ B : |x| < 1}, we have −1 −1 ψ (mB) ⊂ S, so ψ (oB)/S is an almost zero R-module. Given y ∈ S/(p), choose a lift y ∈ S of y. Since B satisfies (ii) and ψ(B[p−1]) is dense in S, we −1 p −1 can find z ∈ ψ (oB) for which u := z − y satisfies |ψ(u)| ≤ p . In particular, −1 p 2 u ∈ ψ (mB) ⊂ S; moreover, we may write x1 = −p + p w for some w ∈ R and then write p 2 p−1 u = p(u/p) = (−x1 + p w)(u/p) = −x1(x1 u/p) + puw. p−1 −1 The quantity x1 u/p again belongs to ψ (mB) ⊂ S, so u ∈ (x1, p)S. Therefore p Frobenius is surjective on S/(x1, p); using the fact that x2 ≡ x1 mod pR, we i deduce that Frobenius is surjective on S/(x1, p) for i = 2, . . . , p. Therefore, S satisfies (xviii); since (xvi) is again evident, S satisfies (ii). This proves (a). The proofs of (b) and (c) similarly reduce to the corresponding statements about oA and oB, for which see [9, Theorem 5.5.9] or [11, Theorem 5.25].

Corollary 5.3. For R and S as in Theorem 5.2, ΩS/R = 0. n Proof. Since Sp is finite étaleover Rp,ΩS/R is killed by p for some nonnegative integer n. If n > 0, then for each x ∈ S we may apply Theorem 5.2 to write x = yp + pz, and so dx = pyp−1 dy + p dz is also killed by pn−1. By induction, it follows that we may take n = 0, proving the claim. Remark 5.4. Parts (b) and (c) of Theorem 5.2 remain true if we replace S with a 0 R-subalgebra S of Sp which is almost equal to S. Remark 5.5. Let R be a Witt-perfect valuation ring in which p 6= 0. Then Theo- rem 5.2 implies that for S the integral closure of R in a finite extension of Frac(R), the maximal ideal of S surjects onto the maximal ideal of R under the trace map. ON THE WITT VECTOR FROBENIUS 15

In other words, R is deeply ramiﬁed in the sense of Coates and Greenberg [1]. The case R = OCp of this result was previously established by Tate. Remark 5.6. The result of Tate described in Remark 5.5 was previously generalized by Faltings’s original almost purity theorem, which may also be deduced from Theorem 5.2. A typical case of the latter result covered by Faltings is ± ± 1/p∞ 1/p∞ R = Zp[µp∞ ][T1 ,...,Tn ][T1 ,...,Tn ], for which it is clear that R is Witt-perfect but not at all apparent that S is. Theorem 5.2 also includes, and indeed is a reformulation of, the generalizations of almost purity given in [9, Theorem 5.5.9] and [11, Theorem 5.25]. Those results are stated in terms of p-adically complete rings and use nonarchimedean analytic geometry in their proofs. The reformulation in terms of the Witt-perfect condition suggests the intriguing possibility of looking at Witt-perfectness for multiple primes at once, with a view towards extending the constructions of p-adic Hodge theory to a more global setting.

Acknowledgements The authors thank Laurent Berger, Bhargav Bhatt, James Borger, Lars Hes- selholt, Abhinav Kumar, Ruochuan Liu, Joe Rabinoﬀ, and Liang Xiao for helpful discussions.

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University of California, Irvine, Dept of Mathematics, Irvine, CA 92697 E-mail address: [email protected]

University of California, San Diego, Dept of Mathematics, La Jolla, CA 92093 E-mail address: [email protected]